In a large bowl, combine the ground chicken, feta cheese, Greek yogurt, breadcrumbs, minced garlic, oregano, lemon juice, salt, black pepper, and red pepper flakes (if using). Mix until just combined, being careful not to overwork the meat.
Divide the mixture into 4 equal portions and shape each portion into a patty about 1/2 inch thick.
Preheat a grill or skillet over medium heat. Lightly grease the grill or skillet with cooking spray or a little olive oil.
Cook the patties for about 5-6 minutes on each side, or until the internal temperature reaches 165°F and the burgers are golden brown.
Toast the burger buns on the grill or in a toaster until lightly browned.
Assemble the burgers by placing a patty on each bun, then topping with baby spinach, sliced tomato, and cucumber. Serve immediately.